Steve314
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 Dec 12 awarded Nice Question Nov 26 awarded Famous Question Sep 22 awarded Famous Question Sep 19 revised Relative sizes of sets of integers and rationals revisited - how do I make sense of this? A hopefully helpful example added May 13 awarded Popular Question Mar 11 awarded Popular Question Dec 18 comment If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? @Bill - it turns out that Euclids Lemma was the bit I was missing for in that inductive proof, though it only really restates the shared-prime-factors proof anyway. Dec 18 comment If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? @Bill - OK, I hadn't thought about it that way. An obvious idea for an inductive proof (that might be more general) has been bugging me, but it seems on the one hand so trivial it's silly, on the other hand to be bugging me that somethings missing. Anyway, I should have time to work through some of these things in the next few days and it's so long since I made time for some math-related fun, so I'll get back to that. Dec 18 comment If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? @Bill - there's simpler things I took as obvious too, such as dealing with negative integer powers. One thing that's been niggling me, though, is why my brain insists on prime factors. Any shared factor between numerator and denominator will do. I guess it's because you can't express a number as a product of all its factors, as MPW showed above, so although it's not necessary, it's a nice form. Also, all factors of a number are products of some subbag (if that's a word) of its prime factors anyway, so in a sense all factors are still covered. Dec 16 accepted If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? Dec 16 awarded Self-Learner Dec 16 awarded Yearling Dec 16 comment If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? OK, that much makes sense, but I'm going to have to re-read this after some sleep. Dec 16 comment If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? Shouldn't $gcd(a,c)$ be smaller than both $a$ and $c$? I can see there should be a proof based on the greatest common divisor, if only because it's equal to the product of the shared prime factors, but something seems odd here. Should some of these GCDs be least common multiples? Dec 16 answered If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? Dec 16 asked If I raise a simplest-terms rational to an integer power, is the result automatically in simplest terms? Dec 8 awarded Caucus Jul 21 answered Why is $\sin(d\Phi) = d\Phi$ where $d\Phi$ is very small? Jul 2 awarded Curious Mar 4 comment Are there more rational numbers than integers? @caters - possibly simpler way to put it - there's at least one way to biject rationals to/from positive integers. It doesn't matter that you can't determine which rational has count n without first checking all the rationals for counts 1..n-1 for possible duplicates. Having a non-recursive way to compute the mapping isn't a requirement.